math192_hw10sp06 - HW10 Solutions 16.1 Line Integrals 1....

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Unformatted text preview: HW10 Solutions 16.1 Line Integrals 1. (c) 2. (e) 3. (g) 4. (a) 5. (d) 6. (b) 7. (f) 8. (h) 10. r ( t ) = t i + (1 t ) j + k , t 1 d r dt = i j | d r dt | = 2; x = t, y = 1 t, z = 1 x y + z = 2 = t (1 t ) + 1 2 = 2 t 2 integraltext c f ( x,y,z ) ds = integraltext 1 (2 t 2) 2 dt = 2[ t 2 2 t ] 1 = 2 12. r ( t ) = (4 cos t ) i + (4 sin t ) j + 3 t k , 2 t 2 d r dt = ( 4 sin t ) i + (4 cos t ) j + 3 k | d r dt | = radicalbig 16 sin 2 t + 16 cos 2 t + 9 = 5 radicalbig x 2 + y 2 = radicalbig 16 cos 2 t + 16 sin 2 t = 4 integraltext c f ( x,y,z ) ds = integraltext 2 - 2 (4)(5) dt = [20 t ] 2 - 2 = 80 16. integraltext C f ( x,y,z ) ds = integraltext C 1 fds + integraltext C 2 fds + integraltext C 3 fds = 1 3 + ( 1 3 ) = 1 2 = 1 6 20. r ( t ) = (1 t ) i + 1 2 (1 t ) 2 j , t 1 | d r dt | = radicalbig 1 + (1 t ) 2 ; f ( x,y ) = f ( (1 t ) , 1 2 (1 t ) 2 ) = (1- t )+ 1 4 (1- t ) 4 1+(1- t ) 2 integraltext C fds = integraltext 1 (1- t )+ 1 4 (1- t ) 4 1+(1- t ) 2 radicalbig 1 + (1 t ) 2 dt = integraltext 1 ( (1 t ) + 1 4 (1 t ) 4 ) dt = bracketleftbig 1 2 (1 t ) 2 1 20 (1 t ) 5 bracketrightbig 1 = 0 ( 1 2 1 20 ) = 11 20 24. r ( t...
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This note was uploaded on 02/05/2008 for the course MATH 1920 taught by Professor Pantano during the Spring '06 term at Cornell University (Engineering School).

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math192_hw10sp06 - HW10 Solutions 16.1 Line Integrals 1....

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